 So for the purposes of today, it will be enough to start with a single vector space, V over some field, which will usually be the complex numbers or some rational functions over the complex numbers. And in our matrix will be some rational value and a morphism of the tensor square of V. Or another way we can think about this is by putting each vector space into some family of evaluation representations. So I just tensor width with fields over some formal parameter. And we want to think of the R matrix as some matrix depending on the difference of two evaluation parameters. And an R matrix will be a solution to the Yang-Baxter equation. And so this Yang-Baxter equation will consist of three R matrices composed in the opposite order. So first, the three R matrices like this. And here, R1, 2, let's say this will be some equality of operators that acts on a tensor product of three copies of this vector space. And here, this acts in just the first two tensor factors. And similarly, R1, 3 acts on the first and third tensor factor and leaves the second tensor factor alone and so on. And so I can compose these three operators. Or I can compose them in the opposite order. And the Yang-Baxter equation tells me that these should be the same operators being on this triple tensor product. And so pictorially, this can be drawn as follows. I have my first two vector spaces interacting. Then I have my second and third. And then I have my first and third. Or I have these interactions in the opposite order. So this is this picture of Reitermeister 3. So if v is finite dimensional, this is a pretty hard equation to solve. There's the dimension of v to the sixth power equations and then v to the fourth power unknowns. And so if you have a solution, that's a pretty powerful thing. Here's an example of a solution. So I could take my field kappa to be kappa k to be rational functions in some variable kappa. And one example would be the following, acting on, say, the tensor product of C2 with itself. And so it's an exercise, not a hard one, to check that this indeed satisfies the Yang-Baxter equation. And when our matrix will just be a solution to the Yang-Baxter equation. So algebraically, these arise as the non-commutativity constraints of quantum group representations. And so if I have a quantum group to it, I can associate an R matrix that tells me the failure of the tensor product of representations of this quantum group to be non-commutative. The construction also works the other way. So if I have an R matrix, then from this R matrix I can produce a quantum group. So in this way, I start with the representations in some way to braid them. And from there, I produce a quantum group. And so that's called the quantum inverse scattering method. Starting with some R matrix, quantum inverse scattering method. The Fidea of Russia-Tincon produces a certain Hopf algebra, a quantum group, Y called the Yangian. And this quantum group acts on arbitrary tensor products of this representation. And so let me briefly review how this algebra comes about. Give some very, very quick sketch. So generators of Y can be read off from matrix elements of R. So more specifically, these are operators appearing in matrix coefficients of R or compositions of this R matrix. So for example, if I have some endomorphism of V, then the generators can be read off as coefficients of the following operator. I take the trace over V of V applied to the first factor of some long tensor product of V. And then I apply this R matrix however many times I like. And so this in here is some operator that acts on V, V of U, tensor S V to the UI, tensor N copies of this V. If I take the trace over this first factor, or I guess I've called it the zero with factor, then I get some matrix valued series in U. And I want to take the coefficients in, suppose I normalize these R matrices so that R of infinity is 1, then I want to take the coefficients in the 1 over U expansion of this operator here. And these will give me the generators in my Yangian. So again, this formula is not particularly important to internalize right now. I just say it to give you some sense of how this algebra comes about. Some relations among these generators, not all of them, but some can be read off from the Yang-Baxter equation. For those familiar, these are these called the RTT relations. And the resulting algebras are familiar in Schopen other contexts. For example, for this example I wrote down earlier, for this Yang-R matrix that was 1 minus kappa over U, 1, 2. The resulting Yangian is what's called the Yangian of gl2. And so this is some hop-algebra deformation of the universal enveloping algebra of gl2-valued polynomials. And these are studied from many different perspectives. For example, one can write down explicit generators and relations. This was first introduced by Drinfeld, studied by many people, Charlie Pressley, Eddinghoff, Schedler-Schiffman, Moll of Nazarapalshansky. There's lots of work on these Yangians. So from the R-matrix, one can not only read off this algebra, but other special properties of this algebra. So for example, the R-matrix also gives rise to families of commutative subalgebras. And these are called Baxter subalgebras. So if I have some operator in my vector space that commutes with the R-matrix in the following sense, then the resulting operators I get from doing this construction end up commuting. So it's a kind of fun exercise to show that the resulting operators commute. And so in this way, I produce a family of commutative subalgebras for every such operator, phi. So for example, of this Yang-R matrix, I can take this matrix phi to be multiplication by any element of GL2. And to this G, I would associate some commutative subalgebra of this Yangian. So this is the general algebraic package one can produce given these R-matrices. And we'll talk about today how to produce this algebraic package using geometry. And so this, yeah, does it depend on the choice of phi? So one can pick, one picks any, at least here, one picks any phi of finite rank. And all of these phi produce generators that go into the Yangian. So the generators, so I pick all my phi. For each phi, I get some generators. And all of these generators appear in the Yangian. So, yeah, the answer is no. And so this line of study was initiated by Malekin Kunkow, who produced R-matrices from the geometry of Nakajima-Quibber varieties. So this is a pretty extensive theory. Instead of going through it in all this generality, let me just focus on a few examples. So here are some examples of Nakajima-Quibber varieties. If I start with just a quiver that's a point, no loops or anything, and I frame it, then the corresponding quiver variety, I'll turn it around here, I apologize. And the corresponding quiver variety is the cotangent space to the grass manian of KM. And so if, in particular, I take N to be 1, then, well, the comology of X is just what? It's the comology of two points. And I'll set this to be my V. And so if I'm working over C, this just is this C2, which we've already seen. And how do I get these tensor products geometrically? I just take larger N. And so if I take the quiverian comology for some fixed N as K varies, then this turns out to be isomorphic to the tensor to an n-fold tensor product. And here, these evaluation parameters correspond to the quiverian parameters of the torus that acts on the grass manian. And so this produces this kind of C2 vector space that we've seen in this earlier example of Yang-Zar matrix. But of course, one can do this for more complicated quiver varieties. So here's a slightly more complicated variety. One can take the non-contaminated quiver variety associated with the Jordan quiver. So I double that quiver and frame it. And the corresponding variety is instanton moduli space of torsion-free frame sheaves on P2. And so here R is the rank and is the second churn class of these sheaves. And so in particular, when R is 1, 1 recovers the Hilbert scheme of points that Lothar talked about yesterday. So this is equal to the Hilbert scheme of n points on C2. And one has some torus whose quiverian parameters I'll call T1 and T2 that acts on P2 or C2 here. And that gives rise to an action on the moduli space of sheaves on P2. So it's framed, so the first churn class is automatically 0. And the vector space I want to associate is the union over all Hilbert schemes of C2. And so unlike our previous examples, now V is infinite dimensional. And just like in this picture, I can produce tons of products of V by increasing the size of the framing. And a similar idea holds for an arbitrary Nakajima variety. By increasing the size of the framing, I can produce tons of products of what I get for simple framings. So this V here, I'll say this in a little bit, but this is a Fox space for the Heisenberg algebra. So in this level of generality, Malik and Akunkov construct arm matrices. And they do this for an arbitrary Nakajima variety. And they do it using special Lagrangian correspondences called stable envelopes. So in the interest of time, I'll kind of omit a detailed explanation of these. But let me just say from the geometric properties of these correspondences, the Yang-Baxter equation follows immediately. So most of the work goes into constructing these correspondences. But once you have these correspondences, the Yang-Baxter is pretty much just a formal consequence of properties of these correspondences. And another consequence of the stable envelope construction is that vacuum matrix elements, so certain vacuum matrix elements can be written, so vacuum matrix elements of the corresponding arm matrices can be written in terms of tautological bundles. So let me just say what the tautological bundles are in this example of the Hilbert scheme. So on the Hilbert scheme, there's not so many different choices of tautological bundle. What I associate is given some sub-scheme Z of C2, the fiber should be just sections of the structure shape of this sub-scheme. And this gives rise to a rank and vector bundle over the Hilbert scheme of points of C2. This is what's called the tautological bundle. And given quiver varieties, I have similar bundles, one sitting over every node. So what do I mean by two? For example, if I take this matrix and I restrict it to, say, the vacuum, which I just defined to be the homology of the Hilbert scheme of zero points on C2, which is just a point, a tensor with the homology of the Hilbert scheme of n points on C2, and I expand this matrix element here in powers of 1 over u when I get a sum constant. But then as soon as I go to the 1 over u squared term, the tautological bundles begin to appear. And so what are the Yangians one obtains? So I told you any time you have an R matrix, you get a Yangian. This example of the tangent space to gross monions, this gives rise to this Drinfeld-Yangian of gl2 that acts on homology of the tangent space to gross monions. And so this recovers a construction of Varan-Yolo or Nakajima in the corresponding construction in k theory. But if one instead starts with instanton moduli space, one gets a larger algebra. One gets what can be called the Yangian of gl1 hat. So this is some Hopf-Algebra deformation of double loops in gl1, that is to say, Heisenberg algebra valued polynomials. And this was shown by Schiffman and Vasserot to be isomorphic to the comblogical Hall algebra associated to the same quiver. And in general, starting with any quiverity, one gets some Yangian. And especially in affiner or wild type, these algebras are quite big. But they turn out to be just the right size for innumerative applications. So I mentioned not only does one get a Yangian action from these r matrices, but if you can find operators that commute with the r matrices, then one gets Baxter subalgebras. And so let me focus on the example of the Hilbert scheme. So given some complex number q, I can define an operator phi of q that acts on the co-immology of the Hilbert scheme of n points in C2 by q raised to the nth power. So it just counts the number of points and applies q raised to that power. And then it's not hard to check that this r matrix preserves the total number of points appearing in a tensor product, which implies that this phi of q transferred with itself commutes with the r matrix. And in this way, one associates some Baxter subalgebra B of q inside this Yangian. So we have some Baxter subalgebra, this algebraic object. And just like we've found every other algebraic object geometrically, it'd be nice to have a geometric description of this. And indeed we do. So it's a theorem of Malik and Akunkov that these Baxter subalgebras, B of q, can be identified with the algebra of operators of quantum multiplication in the co-immology of instanton moduli space. And so this identification gets one a large part of the way towards writing down explicit formulas for quantum multiplication in co-immology of these instanton moduli spaces. And so not only have we produced all this algebra geometrically, but it turns out that this algebra seems to be kind of exactly the right setting in which to study the enumerative geometry of quiver variety. So that's the kind of general picture. And now I want to see if we can push it a little bit beyond the setting of quiver varieties. So question, what part of this story, a good place to start, is instead of working with just Hilbert schemes of points on C2 or ADE surface resolutions, which are quiver varieties, what if instead I look at the Hilbert scheme of points on a general surface? So there's some motivation for this question. So some motivation. Well, one are conjectural formulas over Dieck, for Dieck and Pixton, for the quantum co-immology of the Hilbert scheme of points on a K3 surface. So maybe if we can reproduce this algebraic package for an arbitrary surface, that can help us pin down what these formulas should be. Another motivation, and I suspect this will be similar kind of ideas will show up in Gertsch's talk and Laraker's talk, are to better understand universal structures for modulite schemes on surfaces, so in particular the Hilbert scheme of points on a surface. So one will see that there are many structures on Hilbert schemes that don't depend so sensitively on the surface, but rather just on its churn numbers and other kind of numerical data of the associated geometry. The third reason why one might care, which I won't write down, but there's expected to be, especially in the case of K-trivial surfaces, almost no wall crossing between the co-immology of the Hilbert scheme of points on a surface and the d-t or p-t theory of three folds that are fibrid in this surface over a curve. And so this whole package can't be reproduced for arbitrary s. And in general, it seems unreasonable to expect that for the Hilbert scheme of points on a surface, one has some explicit identification between any backs or subalgebra and any operator of quantum multiplication. So the whole package can't be reproduced for general s. One, one still can construct an r-matrix. And two, the matrix elements of this r-matrix still encode in terms of, they still encode multiplication by tautological bundles. We were in terms of tautological bundles. And in fact, one can modify this construction to get, so if you have a quiver variety, you have tautological bundles associated, and you have a surface with more bundles on it, then one has to any bundle on the surface an associated tautological bundle. And one can modify this construction to get churn classes of those tautological bundles as well. So that is to say that the classical co-immology of the Hilbert scheme can be identified with a q equals 0 backster subalgebra associated to the Hilbert scheme of s. So yeah, what I'd like to explain is how to construct this r-matrix before I do that. Are there any questions? So in what generality can we do this? So I want to start with some smooth quasi-projective surface. And I need to impose one assumption. So either we need s to be proper. Or I suppose this first thing in the first case is a subcase of what I'm about to say. But where we need a torus to act on s, such that the fixed locus is proper. And so the reason they need to impose this assumption is so we can make sense of what it means to integrate over the surface. And so then the equivariant co-immology of the surface, and I'll suppress dependence on t from now on, has the structure of Frobenius algebra. That is there's some pairing for co-immology classes on the surface. We can just integrate the product of co-immology classes. And if this s is non-proper, this integral is defined as some equivariant residue. But that's fine. And so this is some algebra over equivariant co-immology of a point suitably localized. You may as well invert everything. And so if we have some torus action on s, then the torus also acts on the Hilbert Scheme with points on s. And just as before, we can group all the co-immology of the Hilbert Schemes to obtain some vector space. And this will be a Fox space for a Heisenberg algebra labeled by elements of the co-immology of s. And so how does one construct this Heisenberg algebra? It's constructed by certain correspondences. So just to fix the notation, in this triple product of two Hilbert Schemes, I apologize. And the surface itself, I can associate the following locus. So given some integers m and some positive integer n, I can look at the locus of sub-schemes, such that one is contained in the other. And the difference in support of the two sub-schemes is contained entirely at a point. And then this triple product comes with three projection maps, one into each factor. And I can define an operator going from the co-immology of the left to the co-immology of the right as follows. So I take in some co-immology class x. I pull it back along this map p1. Then I take the co-immology class gamma that I've associated here gamma, I should say, as a co-immology class of my surface. I pull back that class. Then I intersect with the class of this locus y I've defined. And finally, I push the whole thing forward along this map p3. Informally, all I've done is add a thick point of length n along the Poincare dual to this class gamma. And then for these operators, they can be defined analogously or as adjoints to the operators I've already defined. And so these operators are called creation operators. These alpha negative n and these alpha n are called annihilation operators. And so as their notation may suggest, it's a result of the Nakajima independently Grinowski that these operators form a copy of the Heisenberg algebra with labels in the co-immology of my surface. That is to say that the super commutator of two of these operators is 0 if i plus j isn't 0, otherwise it's given by their pair. And then vs can be thought of as a Fox space for the algebra generated by these. So I have generators for all n positive and all n negative. But what happens when n is equal to 0? So what about these 0 modes? Well, I can just add in some central element for each co-immology class. And we can say they act by 0 on the Fox space. But this also gives us these 0 modes also give us an extra degree of freedom. So we can put vs into a family of evaluation representations as we did before, vsu, where here u is some formal parameter. And now these 0 modes will account for the appearance of this u. So that is to say this a0 of gamma will act by a scalar u times the integral of this gamma. And it's convenient to include this minus sign. So that lets us deform this family of representations. So anytime we have a Heisenberg algebra, one can produce, via the Fagan-Fuss construction, a VeroSoro algebra. So these VeroSoro generators can just be written as quadratic expressions in the Heisenberg generators. And so how do we do that? Well, first, it'll again be convenient to, as we were earlier, introduce another formal parameter. Let me, before giving an explicit version of this VeroSoro, let's see how the VeroSoro will arise geometrically. So let me drop this for now and come back to it in a bit. So now what we want to do, so keeping that in mind, what we want to do is construct an R matrix acting in the tensor product of two of these Fox bases labeled by two evaluation parameters. So how do we come up with this? Well, we look at the construction in the case of C2 and see if there's a way we can characterize it in terms of algebra. And so in a special case, when S is equal to C2, the stable envelope construction implies that this RU commutes with all operators that look as follows. If I take a Heisenberg tensor of the unit and that unit tends for the Heisenberg acting on this tensor product, I see that RU commutes with these. And so to that end, let me try to generalize that feature to arbitrary surfaces. What I do is I set alpha plus n to be alpha minus n gamma tensor 1 plus 1 tensor alpha minus n gamma. And similarly, it's minus operators. And I can decompose this tensor product in terms of these two types of operators. That is, I also set Vs plus to be whatever field I'm working over to be the collection of tensors obtained by applying these plus operators to the vacuum tensor itself. And Vs minus is defined similarly. And so one can decompose this tensor product in terms of Vs plus and Vs minus. And once one keeps track of what happens to the evaluation parameters, it gets the following. And so this statement here that RU commutes with all of these alpha plus implies that when s is equal to c2, R of U restricted to this plus part. The resulting R matrix restricted to this plus part is just the identity. And so if we're going to try to generalize this construction to an arbitrary surface, we may as well force it to act just in the second tensor factor here. And so the next question is, how does on the second tensor factor? And again, this has some answer that can be generalized. And the answer will be in terms of this promised V-serial algebra when s is equal to c2. And this can be generalized. And so what's the answer? Now let me write down the V-serial algebra in terms of these generators, in terms of these Heisenberg generators alpha minus. So now I'm going to introduce a new parameter. And I'll set l and gamma capital to be the following quadratic expression in the Heisenberg generators. And it'll be a sum over all insurers m of the normally ordered product alpha m minus, alpha n minus m minus, applied to the coproduct of gamma. So we mentioned that the comology of s is a Frobenius algebra. In particular, it has a coproduct. I take the coproduct of gamma, decompose it into tensor factors, and feed the first factor into alpha m minus, and the second factor into alpha n minus m minus, a term that depends on kappa. And finally, this won't be of particular importance, but there's a term that only appears when n is 0. And so this is some algebraic expression. It also arises geometrically when one takes the commutator of these Heisenberg generators with the boundary of the Hilbert scheme, or the first Schoen class of the tautological bundle. And indeed, these do form a copy of the Virussar algebra. If I commute two of these generators, what I get is the usual Virussar relations now with co-immological labels. And so this quantity in here, this Euler class of the surface minus 6 kappa squared, can be thought of as the central charge of this representation. And then this Vs is the lowest weight representation, the Virussaro. And the conformal dimension is obtained by looking at how these zero modes act. So now I guess this is my Vs minus. And it acts by this quantity. So both the central charge and conformal dimension are quadratic functions of kappa. And so in particular, if I send kappa to minus kappa, I get the same answer. And so what happens is one has an isomorphism that I'll call rs minus that goes from Vs minus to Vs minus that just sends some product of these Virussaro with kappa applied to the vacuum to the same exact sequence of generators, only now I replace my kappa by minus kappa. So what I can do is I can set my r matrix to be the identity on this Vs plus part tensored with rs minus. And this will be some operator acting on the tensor products of these representations. So in the remaining two minutes, let me state the punch line. So starting from this intertwiner of Virussaro representations, we've obtained some r matrix. And indeed, the theorem is that this r matrix satisfies the Ang-Baxter equation. And so the proof goes by reducing this argument for arbitrary s to the case of s2, or to the case of c2 rather, and then using the fact that we have the stable basis construction for c2. So for c2, you have to match up this construction with a stable basis. And then from there, you can build up to general s. And then the second result is that this construction can be modified so that matrix elements, RU, as acting on the homology of the Hilbert scheme, can be written in terms of tautological limits. So let's say given a line bundle L on s, the construction can be modified to produce an r matrix taking its input, this line bundle. And given that line bundle, we can get matrix elements of this tautological bundle. And I just want to end by saying some open questions. So one, is there an independent reason for this connection between the Virussaro algebra and Yang-Baxter? So is there some alternative explanation? So if I do this Virussaro construction for c2, I happen to get an r matrix that matches up with what I get from this other stable envelope construction. And I know that from that stable envelope construction that I have Yang-Baxter. But it would be very nice if there were some alternative or more higher level explanation as to why one would expect Yang-Baxter to be lurking inside this Virussaro algebra. I don't know. And then a second question is what happens if I replace the co-homology of my surface by the co-homology of some higher dimensional projective variety or some other topological space? All we needed, the only input used to define this Virussaro algebra was the Frobenius algebra structure and co-homology of the surface. And so in principle, I could start with the co-homology of some larger projective variety or some larger variety with a torus action, run the same construction, and still ask at the end if I satisfy Yang-Baxter. Yeah, it would be interesting if this were some special feature of surfaces or if it should hold for more general classes of Frobenius algebras. And so I need it even replaced by co-homology of some variety. I could also replace by an arbitrary graded skew symmetric Frobenius algebra. And I could still define the Heisenberg. I could still define the Virussaro and ask whether I would get satisfies Yang-Baxter. So yeah, I'll stop there. Thanks for your attention.